Georgia Institute of Technology College of Sciences

One essential challenge in the computational modeling of multiscale systems is the availability of reliable and interpretable closures that faithfully encode the micro-dynamics. For systems without clear scale separation, there generally exists no such a simple set of macro-scale field variables that allow us to project and predict the dynamics in a self-determined way. We introduce a machine-learning (ML) based approach that enables us to reduce high-dimensional multi-scale systems to reliable macro-scale models with low-dimensional variational structures that preserve canonical degeneracies and symmetry constraints. The non-Newtonian hydrodynamics of polymeric fluids is used as an example to illustrate the essential idea. Unlike our conventional wisdom about ML modeling that focuses on learning the PDE form, the present approach directly learns the energy variational structure from the micro-model through an end-to-end process via the joint learning of a set of micro-macro encoder functions. The final model, named the deep non-Newtonian model (DeePN2), retains a multi-scale nature with clear physical interpretation and strictly preserves the frame-indifference constraints. We show that DeePN2 can capture the broadly overlooked viscoelastic differences arising from the specific molecular structural mechanics without human intervention.